Chair of Statistical Field Theory, École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院统计场论主席）；Laboratory of Computational Neuroscience, École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院计算神经科学实验室）
Arthur Jacot, Berfin Şimşek, Francesco Spadaro, Clément Hongler, Franck Gabriel
发表信息：
Proceedings of the International Conference on Machine Learning, 2020, pp. 7397-7406
Random Feature (RF) models are used as efficient parametric approximations of kernel methods. We investigate, by means of random matrix theory, the connection between Gaussian RF models and Kernel Ridge Regression (KRR). For a Gaussian RF model with $P$ features, $N$ data points, and a ridge $λ$, we show that the average (i.e. expected) RF predictor is close to a KRR predictor with an effective ridge $\tildeλ$. We show that $\tildeλ > λ$ and $\tildeλ \searrow λ$ monotonically as $P$ grows, thus revealing the implicit regularization effect of finite RF sampling. We then compare the risk (i.e. test error) of the $\tildeλ$-KRR predictor with the average risk of the $λ$-RF predictor and obtain a precise and explicit bound on their difference. Finally, we empirically find an extremely good agreement between the test errors of the average $λ$-RF predictor and $\tildeλ$-KRR predictor.
By using the framework of Determinantal Point Processes (DPPs), some theoretical results concerning the interplay between diversity and regularization can be obtained. In this paper we show that sampling subsets with kDPPs results in implicit regularization in the context of ridgeless Kernel Regression. Furthermore, we leverage the common setup of state-of-the-art DPP algorithms to sample multiple small subsets and use them in an ensemble of ridgeless regressions. Our first empirical results indicate that ensemble of ridgeless regressors can be interesting to use for datasets including redunda…
Deep (neural) networks have been applied productively in a wide range of supervised and unsupervised learning tasks. Unlike classical machine learning algorithms, deep networks typically operate in the \emph{overparameterized} regime, where the number of parameters is larger than the number of training data points. Consequently, understanding the generalization properties and the role of (explicit or implicit) regularization in these networks is of great importance. In this work, we explore how the oft-used heuristic of \emph{data augmentation} imposes an {\em implicit regularization} penalty …
Stanford University（斯坦福大学）；Purdue University（普渡大学）；Institute for Advanced Study（高等研究院）
We consider networks, trained via stochastic gradient descent to minimize $\ell_2$ loss, with the training labels perturbed by independent noise at each iteration. We characterize the behavior of the training dynamics near any parameter vector that achieves zero training error, in terms of an implicit regularization term corresponding to the sum over the data points, of the squared $\ell_2$ norm of the gradient of the model with respect to the parameter vector, evaluated at each data point. This holds for networks of any connectivity, width, depth, and choice of activation function. We interpr…
Graduate School of Informatics, Kyoto University（京都大学信息学研究生院）；School of Informatics and Mathematical Science, Faculty of Engineering, Kyoto University（京都大学工学部信息与数学科学学院）
Over-parameterized neural networks generalize well in practice without any explicit regularization. Although it has not been proven yet, empirical evidence suggests that implicit regularization plays a crucial role in deep learning and prevents the network from overfitting. In this work, we introduce the gradient gap deviation and the gradient deflection as statistical measures corresponding to the network curvature and the Hessian matrix to analyze variations of network derivatives with respect to input parameters, and investigate how implicit regularization works in ReLU neural networks from…
Stanford University（斯坦福大学）；Microsoft Research（微软研究院）；University of Washington（华盛顿大学）
Dropout is a widely-used regularization technique, often required to obtain state-of-the-art for a number of architectures. This work demonstrates that dropout introduces two distinct but entangled regularization effects: an explicit effect (also studied in prior work) which occurs since dropout modifies the expected training objective, and, perhaps surprisingly, an additional implicit effect from the stochasticity in the dropout training update. This implicit regularization effect is analogous to the effect of stochasticity in small mini-batch stochastic gradient descent. We disentangle these…
In machine learning and statistical data analysis, we often run into objective function that is a summation: the number of terms in the summation possibly is equal to the sample size, which can be enormous. In such a setting, the stochastic mirror descent (SMD) algorithm is a numerically efficient method -- each iteration involving a very small subset of the data. The variance reduction version of SMD (VRSMD) can further improve SMD by inducing faster convergence. On the other hand, algorithms such as gradient descent and stochastic gradient descent have the implicit regularization property th…
Tel Aviv University（特拉维夫大学）；Google Research（谷歌研究院）
The notion of implicit bias, or implicit regularization, has been suggested as a means to explain the surprising generalization ability of modern-days overparameterized learning algorithms. This notion refers to the tendency of the optimization algorithm towards a certain structured solution that often generalizes well. Recently, several papers have studied implicit regularization and were able to identify this phenomenon in various scenarios. We revisit this paradigm in arguably the simplest non-trivial setup, and study the implicit bias of Stochastic Gradient Descent (SGD) in the context of …
For infinitesimal learning rates, stochastic gradient descent (SGD) follows the path of gradient flow on the full batch loss function. However moderately large learning rates can achieve higher test accuracies, and this generalization benefit is not explained by convergence bounds, since the learning rate which maximizes test accuracy is often larger than the learning rate which minimizes training loss. To interpret this phenomenon we prove that for SGD with random shuffling, the mean SGD iterate also stays close to the path of gradient flow if the learning rate is small and finite, but on a m…
A major obstacle to achieving global convergence in distributed and federated learning is the misalignment of gradients across clients, or mini-batches due to heterogeneity and stochasticity of the distributed data. In this work, we show that data heterogeneity can in fact be exploited to improve generalization performance through implicit regularization. One way to alleviate the effects of heterogeneity is to encourage the alignment of gradients across different clients throughout training. Our analysis reveals that this goal can be accomplished by utilizing the right optimization method that…
Kernel methods have achieved very good performance on large scale regression and classification problems, by using the Nyström method and preconditioning techniques. The Nyström approximation -- based on a subset of landmarks -- gives a low rank approximation of the kernel matrix, and is known to provide a form of implicit regularization. We further elaborate on the impact of sampling diverse landmarks for constructing the Nyström approximation in supervised as well as unsupervised kernel methods. By using Determinantal Point Processes for sampling, we obtain additional theoretical results con…
Conflicting Biases at the Edge of Stability: Norm versus Sharpness Regularization
Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）；Prusa Research（普拉萨研究公司）；Institute for Robotics and Mechatronics, DLR-German Aerospace Center（德国航空航天中心机器人与机电研究所）
Ensemble methods that average over a collection of independent predictors that are each limited to a subsampling of both the examples and features of the training data command a significant presence in machine learning, such as the ever-popular random forest, yet the nature of the subsampling effect, particularly of the features, is not well understood. We study the case of an ensemble of linear predictors, where each individual predictor is fit using ordinary least squares on a random submatrix of the data matrix. We show that, under standard Gaussianity assumptions, when the number of featur…
Komaba Institute for Science, the University of Tokyo（东京大学驹场科学研究所）；University of Twente（特文特大学）
The classical statistical learning theory implies that fitting too many parameters leads to overfitting and poor performance. That modern deep neural networks generalize well despite a large number of parameters contradicts this finding and constitutes a major unsolved problem towards explaining the success of deep learning. While previous work focuses on the implicit regularization induced by stochastic gradient descent (SGD), we study here how the local geometry of the energy landscape around local minima affects the statistical properties of SGD with Gaussian gradient noise. We argue that u…
Stanford University（斯坦福大学）；University of Pennsylvania（宾夕法尼亚大学）；Carnegie Mellon University（卡内基梅隆大学）
We study the implicit regularization of mini-batch stochastic gradient descent, when applied to the fundamental problem of least squares regression. We leverage a continuous-time stochastic differential equation having the same moments as stochastic gradient descent, which we call stochastic gradient flow. We give a bound on the excess risk of stochastic gradient flow at time $t$, over ridge regression with tuning parameter $λ= 1/t$. The bound may be computed from explicit constants (e.g., the mini-batch size, step size, number of iterations), revealing precisely how these quantities drive the…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Johns Hopkins University（约翰斯·霍普金斯大学）；Amazon（亚马逊）；University of Washington（华盛顿大学）；Microsoft Research（微软研究院）
Stochastic gradient descent (SGD) exhibits strong algorithmic regularization effects in practice, which has been hypothesized to play an important role in the generalization of modern machine learning approaches. In this work, we seek to understand these issues in the simpler setting of linear regression (including both underparameterized and overparameterized regimes), where our goal is to make sharp instance-based comparisons of the implicit regularization afforded by (unregularized) average SGD with the explicit regularization of ridge regression. For a broad class of least squares problem …
Stochastic gradient descent (SGD) is a pillar of modern machine learning, serving as the go-to optimization algorithm for a diverse array of problems. While the empirical success of SGD is often attributed to its computational efficiency and favorable generalization behavior, neither effect is well understood and disentangling them remains an open problem. Even in the simple setting of convex quadratic problems, worst-case analyses give an asymptotic convergence rate for SGD that is no better than full-batch gradient descent (GD), and the purported implicit regularization effects of SGD lack a…
University of California, Berkeley（加州大学伯克利分校）；ICSI（国际计算机科学研究所）
Double descent refers to the phase transition that is exhibited by the generalization error of unregularized learning models when varying the ratio between the number of parameters and the number of training samples. The recent success of highly over-parameterized machine learning models such as deep neural networks has motivated a theoretical analysis of the double descent phenomenon in classical models such as linear regression which can also generalize well in the over-parameterized regime. We provide the first exact non-asymptotic expressions for double descent of the minimum norm linear e…
California Institute of Technology（加州理工学院）；Department of Computing and Mathematical Sciences（计算与数学科学系）；Department of Electrical Engineering（电气工程系）
Stochastic descent methods (of the gradient and mirror varieties) have become increasingly popular in optimization. In fact, it is now widely recognized that the success of deep learning is not only due to the special deep architecture of the models, but also due to the behavior of the stochastic descent methods used, which play a key role in reaching "good" solutions that generalize well to unseen data. In an attempt to shed some light on why this is the case, we revisit some minimax properties of stochastic gradient descent (SGD) for the square loss of linear models---originally developed in…
This article is an overview of supervised machine learning problems for regression and classification. Topics include: kernel methods, training by stochastic gradient descent, deep learning architecture, losses for classification, statistical learning theory, and dimension independent generalization bounds. Implicit regularization in deep learning examples are presented, including data augmentation, adversarial training, and additive noise. These methods are reframed as explicit gradient regularization.
In this paper, we revisit implicit regularization from the ground up using notions from dynamical systems and invariant subspaces of Morse functions. The key contributions are a new criterion for implicit regularization---a leading contender to explain the generalization power of deep models such as neural networks---and a general blueprint to study it. We apply these techniques to settle a conjecture on implicit regularization in matrix factorization.
Understanding the implicit regularization (or implicit bias) of gradient descent has recently been a very active research area. However, the implicit regularization in nonlinear neural networks is still poorly understood, especially for regression losses such as the square loss. Perhaps surprisingly, we prove that even for a single ReLU neuron, it is impossible to characterize the implicit regularization with the square loss by any explicit function of the model parameters (although on the positive side, we show it can be characterized approximately). For one hidden-layer networks, we prove a …
National University of Defense Technology（国防科技大学）；University of Utah（犹他大学）
The explicit low-rank regularization, e.g., nuclear norm regularization, has been widely used in imaging sciences. However, it has been found that implicit regularization outperforms explicit ones in various image processing tasks. Another issue is that the fixed explicit regularization limits the applicability to broad images since different images favor different features captured by different explicit regularizations. As such, this paper proposes a new adaptive and implicit low-rank regularization that captures the low-rank prior dynamically from the training data. The core of our new adapt…
Gradient descent can be surprisingly good at optimizing deep neural networks without overfitting and without explicit regularization. We find that the discrete steps of gradient descent implicitly regularize models by penalizing gradient descent trajectories that have large loss gradients. We call this Implicit Gradient Regularization (IGR) and we use backward error analysis to calculate the size of this regularization. We confirm empirically that implicit gradient regularization biases gradient descent toward flat minima, where test errors are small and solutions are robust to noisy parameter…
We study implicit regularization when optimizing an underdetermined quadratic objective over a matrix $X$ with gradient descent on a factorization of $X$. We conjecture and provide empirical and theoretical evidence that with small enough step sizes and initialization close enough to the origin, gradient descent on a full dimensional factorization converges to the minimum nuclear norm solution.
Texas A&M University（德克萨斯农工大学）；New York University（纽约大学）
We study the implicit regularization of gradient descent towards structured sparsity via a novel neural reparameterization, which we call a diagonally grouped linear neural network. We show the following intriguing property of our reparameterization: gradient descent over the squared regression loss, without any explicit regularization, biases towards solutions with a group sparsity structure. In contrast to many existing works in understanding implicit regularization, we prove that our training trajectory cannot be simulated by mirror descent. We analyze the gradient dynamics of the correspon…
Mila, Université de Montréal（蒙特利尔大学米拉研究所）；Microsoft Research（微软研究院）；Canada CIFAR AI chair（加拿大高级研究所加拿大人工智能主席项目）
We approach the problem of implicit regularization in deep learning from a geometrical viewpoint. We highlight a regularization effect induced by a dynamical alignment of the neural tangent features introduced by Jacot et al, along a small number of task-relevant directions. This can be interpreted as a combined mechanism of feature selection and compression. By extrapolating a new analysis of Rademacher complexity bounds for linear models, we motivate and study a heuristic complexity measure that captures this phenomenon, in terms of sequences of tangent kernel classes along optimization path…
Massachusetts Institute of Technology（麻省理工学院）；Istituto Italiano di Tecnologia（意大利技术研究院）；Università degli Studi di Genova（热那亚大学）
We study the generalization properties of stochastic gradient methods for learning with convex loss functions and linearly parameterized functions. We show that, in the absence of penalizations or constraints, the stability and approximation properties of the algorithm can be controlled by tuning either the step-size or the number of passes over the data. In this view, these parameters can be seen to control a form of implicit regularization. Numerical results complement the theoretical findings.
Calculation Consulting（Calculation Consulting公司）；ICSI（国际计算机科学研究所）；University of California at Berkeley（加州大学伯克利分校）
Random Matrix Theory (RMT) is applied to analyze weight matrices of Deep Neural Networks (DNNs), including both production quality, pre-trained models such as AlexNet and Inception, and smaller models trained from scratch, such as LeNet5 and a miniature-AlexNet. Empirical and theoretical results clearly indicate that the DNN training process itself implicitly implements a form of Self-Regularization. The empirical spectral density (ESD) of DNN layer matrices displays signatures of traditionally-regularized statistical models, even in the absence of exogenously specifying traditional forms of e…
Northeast Normal University（东北师范大学）；Beihua University（北华大学）
The study on the implicit regularization induced by gradient-based optimization is a longstanding pursuit. In the present paper, we characterize the implicit regularization of momentum gradient descent (MGD) with early stopping by comparing with the explicit $\ell_2$-regularization (ridge). In details, we study MGD in the continuous-time view, so-called momentum gradient flow (MGF), and show that its tendency is closer to ridge than the gradient descent (GD) [Ali et al., 2019] for least squares regression. Moreover, we prove that, under the calibration $t=\sqrt{2/λ}$, where $t$ is the time par…
Texas A&M University（德克萨斯A&M大学）；AWS AI；New York University（纽约大学）
In this paper, we study the implicit bias of gradient descent for sparse regression. We extend results on regression with quadratic parametrization, which amounts to depth-2 diagonal linear networks, to more general depth-N networks, under more realistic settings of noise and correlated designs. We show that early stopping is crucial for gradient descent to converge to a sparse model, a phenomenon that we call implicit sparse regularization. This result is in sharp contrast to known results for noiseless and uncorrelated-design cases. We characterize the impact of depth and early stopping and …
Mathematically characterizing the implicit regularization induced by gradient-based optimization is a longstanding pursuit in the theory of deep learning. A widespread hope is that a characterization based on minimization of norms may apply, and a standard test-bed for studying this prospect is matrix factorization (matrix completion via linear neural networks). It is an open question whether norms can explain the implicit regularization in matrix factorization. The current paper resolves this open question in the negative, by proving that there exist natural matrix factorization problems on w…
Purdue University（普渡大学）；The University of Chicago（芝加哥大学）；National University of Singapore（新加坡国立大学）
Overfitting frequently occurs in deep learning. In this paper, we propose a novel regularization method called Drop-Activation to reduce overfitting and improve generalization. The key idea is to drop nonlinear activation functions by setting them to be identity functions randomly during training time. During testing, we use a deterministic network with a new activation function to encode the average effect of dropping activations randomly. Our theoretical analyses support the regularization effect of Drop-Activation as implicit parameter reduction and verify its capability to be used together…
In the pursuit of explaining implicit regularization in deep learning, prominent focus was given to matrix and tensor factorizations, which correspond to simplified neural networks. It was shown that these models exhibit an implicit tendency towards low matrix and tensor ranks, respectively. Drawing closer to practical deep learning, the current paper theoretically analyzes the implicit regularization in hierarchical tensor factorization, a model equivalent to certain deep convolutional neural networks. Through a dynamical systems lens, we overcome challenges associated with hierarchy, and est…
Mila Institute（米拉研究所）；Saint Mary’s University（圣玛丽大学）；Université de Montréal（蒙特利尔大学）
We theoretically analyze the Feedback Alignment (FA) algorithm, an efficient alternative to backpropagation for training neural networks. We provide convergence guarantees with rates for deep linear networks for both continuous and discrete dynamics. Additionally, we study incremental learning phenomena for shallow linear networks. Interestingly, certain specific initializations imply that negligible components are learned before the principal ones, thus potentially negatively affecting the effectiveness of such a learning algorithm; a phenomenon we classify as implicit anti-regularization. We…
Recent efforts to unravel the mystery of implicit regularization in deep learning have led to a theoretical focus on matrix factorization -- matrix completion via linear neural network. As a step further towards practical deep learning, we provide the first theoretical analysis of implicit regularization in tensor factorization -- tensor completion via certain type of non-linear neural network. We circumvent the notorious difficulty of tensor problems by adopting a dynamical systems perspective, and characterizing the evolution induced by gradient descent. The characterization suggests a form …
Princeton University（普林斯顿大学）；Institute for Advanced Study（高等研究院）；Tel Aviv University（特拉维夫大学）
Efforts to understand the generalization mystery in deep learning have led to the belief that gradient-based optimization induces a form of implicit regularization, a bias towards models of low "complexity." We study the implicit regularization of gradient descent over deep linear neural networks for matrix completion and sensing, a model referred to as deep matrix factorization. Our first finding, supported by theory and experiments, is that adding depth to a matrix factorization enhances an implicit tendency towards low-rank solutions, oftentimes leading to more accurate recovery. Secondly, …
Boston University（波士顿大学）；Rice University（莱斯大学）；University of Maryland（马里兰大学）
We consider whether algorithmic choices in over-parameterized linear matrix factorization introduce implicit regularization. We focus on noiseless matrix sensing over rank-$r$ positive semi-definite (PSD) matrices in $\mathbb{R}^{n \times n}$, with a sensing mechanism that satisfies restricted isometry properties (RIP). The algorithm we study is \emph{factored gradient descent}, where we model the low-rankness and PSD constraints with the factorization $UU^\top$, for $U \in \mathbb{R}^{n \times r}$. Surprisingly, recent work argues that the choice of $r \leq n$ is not pivotal: even setting $U …
Gradient regularization (GR) is a method that penalizes the gradient norm of the training loss during training. While some studies have reported that GR can improve generalization performance, little attention has been paid to it from the algorithmic perspective, that is, the algorithms of GR that efficiently improve the performance. In this study, we first reveal that a specific finite-difference computation, composed of both gradient ascent and descent steps, reduces the computational cost of GR. Next, we show that the finite-difference computation also works better in the sense of generaliz…
While the impact of variational inference (VI) on posterior inference in a fixed generative model is well-characterized, its role in regularizing a learned generative model when used in variational autoencoders (VAEs) is poorly understood. We study the regularizing effects of variational distributions on learning in generative models from two perspectives. First, we analyze the role that the choice of variational family plays in imparting uniqueness to the learned model by restricting the set of optimal generative models. Second, we study the regularization effect of the variational family on …
University of Pennsylvania（宾夕法尼亚大学）；Stanford University（斯坦福大学）
Acceleration and momentum are the de facto standard in modern applications of machine learning and optimization, yet the bulk of the work on implicit regularization focuses instead on unaccelerated methods. In this paper, we study the statistical risk of the iterates generated by Nesterov's accelerated gradient method and Polyak's heavy ball method, when applied to least squares regression, drawing several connections to explicit penalization. We carry out our analyses in continuous-time, allowing us to make sharper statements than in prior work, and revealing complex interactions between earl…
National Engineering School of Tunis, University of Tunis El Manar（突尼斯埃尔马纳尔大学突尼斯国立工程学院）；Aix Marseille University（艾克斯-马赛大学）；CNRS（法国国家科学研究中心）；Ecole Centrale de Marseille（马赛中央理工学院）
We study the implicit regularization effects of deep learning in tensor factorization. While implicit regularization in deep matrix and 'shallow' tensor factorization via linear and certain type of non-linear neural networks promotes low-rank solutions with at most quadratic growth, we show that its effect in deep tensor factorization grows polynomially with the depth of the network. This provides a remarkably faithful description of the observed experimental behaviour. Using numerical experiments, we demonstrate the benefits of this implicit regularization in yielding a more accurate estimati…
Big Data Lab, Baidu Research（百度研究院大数据实验室）；School of Artificial Intelligence and Automation Huazhong University of Science and Technology（华中科技大学人工智能与自动化学院）
Principal component analysis (PCA) has been widely used as an effective technique for feature extraction and dimension reduction. In the High Dimension Low Sample Size (HDLSS) setting, one may prefer modified principal components, with penalized loadings, and automated penalty selection by implementing model selection among these different models with varying penalties. The earlier work [1, 2] has proposed penalized PCA, indicating the feasibility of model selection in $L_2$- penalized PCA through the solution path of Ridge regression, however, it is extremely time-consuming because of the int…
Deep ReLU networks trained with the square loss have been observed to perform well in classification tasks. We provide here a theoretical justification based on analysis of the associated gradient flow. We show that convergence to a solution with the absolute minimum norm is expected when normalization techniques such as Batch Normalization (BN) or Weight Normalization (WN) are used together with Weight Decay (WD). The main property of the minimizers that bounds their expected error is the norm: we prove that among all the close-to-interpolating solutions, the ones associated with smaller Frob…
Raymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 69978, Israel（雷蒙德与贝弗利·萨克勒物理与天文学学院，特拉维夫大学）；École Polytechnique Fédérale de Lausanne (EPFL), Switzerland（洛桑联邦理工学院（EPFL），瑞士）；Racah Institute of Physics, The Hebrew University of Jerusalem, Edmond J. Safra Campus, Jerusalem 9190401, Israel（拉卡姆物理研究所，耶路撒冷希伯来大学）；Selim Benin School of Computer Science and Engineering, The Hebrew University of Jerusalem, Edmond J. Safra Campus, Jerusalem 9190401, Israel（塞利姆·本因计算机科学与工程学院，耶路撒冷希伯来大学）
We study discrete-time mirror descent applied to the unregularized empirical risk in matrix sensing. In both the general case of rectangular matrices and the particular case of positive semidefinite matrices, a simple potential-based analysis in terms of the Bregman divergence allows us to establish convergence of mirror descent -- with different choices of the mirror maps -- to a matrix that, among all global minimizers of the empirical risk, minimizes a quantity explicitly related to the nuclear norm, the Frobenius norm, and the von Neumann entropy. In both cases, this characterization impli…
